This paper proves a version for stochastic differential equations of the Lie–Scheffers theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution properties of the distribution generated by the vector fields that define it. When stated in the particular case of standard deterministic systems, our main theorem improves various aspects of the classical Lie–Scheffers result. We show that the stochastic analog of the classical Lie–Scheffers systems can be reduced to the study of Lie group valued stochastic Lie–Scheffers systems; those systems, as well as those taking values in homogeneous spaces are studied in detail. The developments of the paper are illustrated with several examples.
Ce papier contient une généralisation du Théorème de Lie–Scheffers aux équations différentielles stochastiques. Ce résultat caractérise l’existence de règles de superposition non linéaires pour la solution générale de ces équations, en termes des propriétés d’involution de la distribution engendrée par les champs vecteurs qui les définissent. Dans le cas particulier des systèmes déterministes, notre théorème principal améliore certains aspects du théorème de Lie–Scheffers traditionnel. Nous montrons que l’analogue stochastique des systèmes de Lie–Scheffers classiques peuvent être réduits à l’étude des systèmes de Lie–Scheffers stochastiques à valeurs dans un groupe de Lie; ces systèmes, ainsi que ceux qui prennent des valeurs dans des espaces homogènes sont étudiés en détail. Les développements de ce papier sont illustrés avec plusieurs exemples.
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